Math 401 - 01 Previous Homework (Fall 2018)
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Problem Set 10 (complete) Due: 11/06/2018. Board presentation: 11/20/2018
Let G be a group, and H,K\leq G.
Prove that if HK=KH, then HK\leq G.
Prove that if H\leq N_G(K), then HK\leq G.
Let G be a group, H\leq G, and C=\{gHg^{-1}|g\in G\} the set of all conjugates of H in G. Prove that: |C|=[G:N_G(H)].
Let G be a group of order 120. What are the possible values of n_2, n_3, and n_5, i.e. the number of Sylow 2-subgroups, the number of Sylow 3-subgroups and the number of Sylow 5-subgroups?
How many groups of order 6727 are there? Describe them. Justify your answers. Show all your work.
Problem Set 09 (complete) Due: 10/29/2018. Board presentation: 11/02/2018
Prove that, up to isomorphism, the direct product operation is commutative and associative.
Give an example of a group G with two subgroups H and K such that HK=G, H\intersection K=1, K\normaleq G, but G is not isomorphic to the direct product H\oplus K.
Let G be a group, and H,N\leq G. Prove that:
If N\normaleq G, then HN\leq G.
If both H,N\normaleq G, then HN\normaleq G.
Make a list of all abelian groups of order 2736. Express each of them using the “elementary divisors” form and the “invariant factors” form.
Problem Set 08 (complete) Due: 10/22/2018. Problem 4 may be resubmitted by 10/24/2018. Board presentation 10/29/2018
Let n\in\N and H\leq G such that H is the only subgroup of G of order n. Show that H\normaleq G. (Do not assume that G is finite)
Let n\in\N and H\leq G such that H is the only subgroup of G of index n. Show that H\normaleq G. (Do not assume that G is finite)
Combine the previous problem with problem 3 in Problem Set 6.
Let p,q be primes such that p < q and p\not\mid (q-1). Prove that, up to isomorphism, there is only one group of order pq. (Hint: Use example 17, page 203, as a guide. No use this example, you may use the extra assumption that (p-1)\not\mid (q-1), or equivalently that (p-1)\not\mid (pq-1).)
Problem Set 07 (complete) Due: 10/15/2018. Board presentation 10/29/2018
Prove Thm. 6.2.3, Thm. 6.3.2, Thm. 10.2.3. Combine all three proofs into one.
Chapter 10, problems 8, 10.
Problem Set 06 (complete) Due: 10/08/2018. Board presentation 10/10/2018
Chapter 7, problem 8.
Chapter 7, problem 22.
Let G be a finite group, and p the smallest prime divisor of |G|. If p^2\not\mid|G|, then G has at most one subgroup of index p. (Hint:Look at Example 6 on page 144)
Chapter 7, problem 12. Generalize.
Chapter 7, problem 48.
Problem Set 05 (complete) Due: 09/24/2018. Board presentation 09/28/2018
Chapter 5, problems 6, 8. For all of them find the order and the parity.
Chapter 5, problem 10. What is the largest order of an element of S_8. Explain.
Chapter 5, problems 23, 24.
Chapter 5, problem 48.
Chapter 5, problem 50. Is D_5 a subgroup of A_5? Explain.
Problem Set 04 (complete) Due: 09/17/2018. Board presentation: 09/24/2018
Chapter 4, problem 74.
Chapter 5, problem 2.a.
Chapter 5, problem 4.
Consider \alpha\in S_8 given in disjoint cycle form by \alpha=(1\ 4\ 5)(3\ 7). Write \alpha in array form.
Problem Set 03 (complete) Due: 09/12/2018. Board presentation: 09/17/2018
Let G=\pbr{a} be an infinite cyclic group, and k_1,k_2\in\Z. Prove that \pbr{a^{k_1}}\leq\pbr{a^{k_2}} \textrm{ iff } k_2\mid k_1.
Let G=\pbr{a} be a cyclic group of order 60.
How many subgroups does G have?
Which of them are cyclic?
List a generator for each of the cyclic subgroups of G.
Draw the subgroup lattice of G.
Prove that a finite group of prime order must be cyclic.
Chap. 4, problem 38, 62.
Chap. 4, problem 50.
Problem Set 02 (complete) Due: 09/04/2018. Board presentation: 09/12/2018
Chap. 3, problems 4, 13, 20, 64
Chap. 3, problems 6, 50
Let G be a group in which every non-identity element has order 2. Prove that G must be Abelian.
Chap. 3, problem 34. what can you say about the union of two subgroups?
Problem Set 01 (complete) Due: 08/27/2018. Board presentation: 08/31/2018
Page 38, prob. 18. What happens if you replace each H with an A?
Page 39, prob. 22. Explain.
Page 54, prob. 4.
Page 56, prob. 22. Compare with problem 13 on page 38.
Additional problems to look at: 1.13 p.38, 1.14 p.38, 2.5 p.54,
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